Appendix A
The operators \(D_{lq} \,(l,q=1,2,3,4)\) are described as:
$$\begin{aligned} D_{11}&=r_{12} h\frac{\partial ^{4}}{\partial \xi ^{4}}+\frac{\left( {r_{11} -r_{31} } \right) h}{\xi ^{2}}\frac{\partial ^{4}}{\partial \xi ^{2}\partial y^{2}}+\frac{\left( {3r_{31} -3r_{11} -r_{21} } \right) h}{\xi ^{3}}\frac{\partial ^{3}}{\partial \xi \partial y^{2}}+\frac{\left( {r_{11} -r_{22} +r_{12} } \right) h}{\xi }\frac{\partial ^{3}}{\partial \xi ^{3}}, \nonumber \\&\quad +\frac{\left( {r_{22} -r_{11} -r_{12} -r_{21} } \right) h}{\xi ^{2}}\frac{\partial ^{2}}{\partial \xi ^{2}}+\frac{3\left( {r_{21} +r_{11} -r_{31} } \right) h}{\xi ^{4}}\frac{\partial ^{2}}{\partial y^{2}}+\frac{2r_{21} h}{\xi ^{3}}\frac{\partial }{\partial \xi }, \nonumber \\ D_{12}&=-r_{13} \frac{\partial ^{4}}{\partial \xi ^{4}}-\frac{r_{14} +r_{32} }{\xi ^{2}}\frac{\partial ^{4}}{\partial \xi ^{2}\partial y^{2}}+\frac{3r_{14} +3r_{32} +r_{24} }{\xi ^{3}}\frac{\partial ^{3}}{\partial \xi \partial y^{2}}-\frac{r_{13} +r_{14} -r_{23} }{\xi }\frac{\partial ^{3}}{\partial \xi ^{3}}, \nonumber \\&\quad +\frac{r_{13} +r_{14} -r_{23} +r_{24} }{\xi ^{2}}\frac{\partial ^{2}}{\partial \xi ^{2}}-\frac{3(r_{14} +r_{24} +r_{32} )}{\xi ^{4}}\frac{\partial ^{2}}{\partial y^{2}}-\frac{2r_{24} }{\xi ^{3}}\frac{\partial }{\partial \xi }, \nonumber \\ D_{13}&=r_{15} \frac{\partial ^{3}}{\partial \xi ^{3}}+\frac{r_{15} -r_{25} }{\xi }\frac{\partial ^{2}}{\partial \xi ^{2}}+\frac{r_{35} }{\xi ^{2}}\frac{\partial ^{3}}{\partial \xi \partial y^{2}}-U_{3} \frac{\partial }{\partial \xi }-\frac{r_{15} -r_{25} }{\xi ^{2}}\frac{\partial }{\partial \xi }-\frac{r_{35} }{\xi ^{3}}\frac{\partial ^{2}}{\partial y^{2}}, \nonumber \\ D_{14}&=\frac{r_{38} +r_{18} }{\xi }\frac{\partial ^{3}}{\partial \xi ^{2}\partial y}-\frac{r_{28} +r_{18} +r_{38} }{\xi ^{2}}\frac{\partial ^{2}}{\partial \xi \partial y}+\frac{2r_{28} }{\xi ^{3}}\frac{\partial }{\partial y}, \nonumber \\ D_{21}&=\frac{r_{21} h}{\xi ^{3}}\frac{\partial ^{4}}{\partial y^{4}}+\frac{\left( {r_{22} -r_{31} } \right) h}{\xi }\frac{\partial ^{4}}{\partial \xi ^{2}\partial y^{2}}+\frac{r_{21} h}{\xi ^{2}}\frac{\partial ^{3}}{\partial \xi \partial y^{2}}, \nonumber \\ D_{22}&=-\frac{r_{32} +r_{23} }{\xi }\frac{\partial ^{4}}{\partial \xi ^{2}\partial y^{2}}-\frac{r_{24} }{\xi ^{3}}\frac{\partial ^{4}}{\partial y^{4}}-\frac{r_{24} }{\xi ^{2}}\frac{\partial ^{3}}{\partial \xi \partial y^{2}}, \nonumber \\ D_{23}&=\frac{r_{25} +r_{35} }{\xi }\frac{\partial ^{3}}{\partial \xi \partial y^{2}}+\frac{r_{35} }{\xi ^{2}}\frac{\partial ^{2}}{\partial y^{2}},\,\,D_{24} =r_{38} \frac{\partial ^{3}}{\partial \xi ^{2}\partial y}+\frac{2r_{38} }{\xi }\frac{\partial ^{2}}{\partial \xi \partial y}+\frac{r_{28} }{\xi ^{2}}\frac{\partial ^{3}}{\partial y^{3}}-U_{2} \frac{\partial }{\partial y}, \nonumber \\ D_{31}&=\frac{l_{11} h}{\xi ^{4}}\frac{\partial ^{4}}{\partial y^{4}}+\frac{\left( {2l_{31} +l_{21} +l_{12} } \right) h}{\xi ^{2}}\frac{\partial ^{4}}{\partial \xi ^{2}\partial y^{2}}-\frac{2(l_{31} +l_{21} )h}{\xi ^{3}}\frac{\partial ^{3}}{\partial \xi \partial y^{2}}, \nonumber \\&\quad +\frac{2(l_{31} +l_{21} +l_{11} )h}{\xi ^{4}}\frac{\partial ^{2}}{\partial y^{2}}+\frac{l_{11} h}{\xi ^{3}}\frac{\partial }{\partial \xi }-\frac{l_{11} h}{\xi ^{2}}\frac{\partial ^{2}}{\partial \xi ^{2}}+\frac{\left( {l_{21} +2l_{22} -l_{12} } \right) h}{\xi }\frac{\partial ^{3}}{\partial \xi ^{3}}+l_{22} h\frac{\partial ^{4}}{\partial \xi ^{4}}\,,\,\, \nonumber \\ D_{32}&=-\frac{l_{14} }{\xi ^{4}}\frac{\partial ^{4}}{\partial y^{4}}+\frac{2l_{32} -l_{13} -l_{24} }{\xi ^{2}}\frac{\partial ^{4}}{\partial \xi ^{2}\partial y^{2}}+\frac{2(l_{24} -l_{32} )}{\xi ^{3}}\frac{\partial ^{3}}{\partial \xi \partial y^{2}}+\frac{2(l_{32} -l_{24} -l_{14} )}{\xi ^{4}}\frac{\partial ^{2}}{\partial y^{2}}, \nonumber \\&\quad -\frac{l_{14} }{\xi ^{3}}\frac{\partial }{\partial \xi }+\left( {\frac{l_{14} }{\xi ^{2}}+\frac{\cot \beta }{\xi }} \right) \frac{\partial ^{2}}{\partial \xi ^{2}}+\frac{l_{13} -l_{24} -2l_{23} }{\xi }\frac{\partial ^{3}}{\partial \xi ^{3}}-l_{23} \frac{\partial ^{4}}{\partial \xi ^{4}}, \nonumber \\ D_{33}&=\frac{2l_{35} +l_{15} }{\xi ^{2}}\frac{\partial ^{3}}{\partial \xi \partial y^{2}}+l_{25} \frac{\partial ^{3}}{\partial \xi { }^{3}}+\frac{2l_{25} -l_{15} }{\xi }\frac{\partial ^{2}}{\partial \xi ^{2}}, \nonumber \\ D_{34}&=\frac{l_{18} }{\xi ^{3}}\frac{\partial ^{3}}{\partial y^{3}}+\frac{2l_{38} +l_{28} }{\xi }\frac{\partial ^{3}}{\partial \xi ^{2}\partial y}+\frac{2l_{38} -l_{18} }{\xi ^{2}}\frac{\partial ^{2}}{\partial \xi \partial y}+\frac{l_{18} }{\xi ^{3}}\frac{\partial }{\partial y}, \nonumber \\ D_{41}&=\frac{h\cot \beta }{\xi }\frac{\partial ^{2}}{\partial \xi ^{2}},\quad D_{42} =-\frac{\xi }{\cot \beta }\left[ {T_{1} \frac{\partial ^{2}}{\partial \xi ^{2}}+T_{2} \left( {\frac{1}{\xi }\frac{\partial ^{2}}{\partial y^{2}}+\frac{\partial }{\partial \xi }} \right) } \right] , \nonumber \\ D_{43}&=U_{3} \left( {\frac{\partial }{\partial \xi }+\frac{1}{\xi }} \right) ,\quad D_{44} =\frac{U_{4} }{\xi }\frac{\partial }{\partial y}. \end{aligned}$$
(A.1)
where
$$\begin{aligned} U_{3}&=\int \limits _{-h_{1} }^{-h_{2} } \;\; {\varsigma _{1}^{(1)} (z)\text { d }z} +\int \limits _{-h_{2} }^{h_{3} } \;\; {\varsigma _{1}^{(2)} (z)\text { d }z} +\int \limits _{h_{3} }^{h_{4} } \;\; {\varsigma _{1}^{(3)} (z)\text { d }z} ,\,\, \nonumber \\ U_{4}&=\int \limits _{-h_{1} }^{-h_{2} } \;\; {\varsigma _{2}^{(1)} (z)\text { d }z} +\int \limits _{-h_{2} }^{h_{3} } \;\; {\varsigma _{2}^{(2)} (z)\text { d }z} +\int \limits _{h_{3} }^{h_{4} } \;\; {\varsigma _{2}^{(3)} (z)\text { d }z} .\, \nonumber \\ r_{11}&=F_{11}^{1} l_{11} +F_{12}^{1} l_{21} ,\,\,r_{12} =F_{11}^{1} l_{12} +F_{12}^{1} l_{21} ,\,r_{13} =F_{11}^{1} l_{13} +F_{12}^{1} l_{23} +F_{11}^{2}, \, \nonumber \\ r_{14}&=F_{11}^{1} l_{14} +F_{12}^{1} l_{24} +F_{12}^{2} ,\,\,r_{15} =F_{11}^{1} l_{15} +F_{12}^{1} l_{25} +F_{15}^{1} ,r_{18} =F_{11}^{1} l_{18} +F_{12}^{1} l_{28} +F_{18}^{1} ,\,\, \nonumber \\ r_{21}&=F_{11}^{1} l_{11} +F_{22}^{1} l_{21} ,\,\,r_{22} =F_{22}^{1} l_{12} +F_{12}^{1} l_{22} ,\,\,r_{23} =F_{21}^{1} l_{13} +F_{22}^{1} l_{23} +F_{21}^{1} ,\,\, \nonumber \\ r_{24}&=F_{22}^{1} l_{14} +F_{22}^{1} l_{24} +F_{22}^{2} ,\,\,\,r_{25} =F_{21}^{1} l_{15} +F_{22}^{1} l_{25} +F_{25}^{1} ,\,\,r_{28} =F_{21}^{1} l_{18} +F_{22}^{1} l_{28} +F_{28}^{1} , \nonumber \\ r_{31}&=F_{66}^{1} l_{31} ,\,\,\,\,r_{32} =F_{66}^{1} l_{32} +2F_{66}^{2} ,\,\,\,\,r_{35} =F_{35}^{1} -F_{66}^{1} l_{35} ,\,\,\,r_{38} =F_{38}^{1} -F_{66}^{1} l_{38} , \nonumber \\ l_{11}&=\frac{F_{22}^{0} }{\Delta };\,\,l_{12} =-\frac{F_{12}^{0} }{\Delta },\,\,\,l_{13} =\frac{F_{12}^{0} F_{21}^{1} -F_{11}^{1} F_{22}^{0} }{\Delta },\,\,\,l_{14} =\frac{F_{12}^{0} F_{22}^{1} -F_{12}^{1} F_{22}^{0} }{\Delta }, \nonumber \\ l_{15}&=\frac{F_{25}^{0} F_{12}^{0} -F_{15}^{0} F_{22}^{0} }{\Delta },\,\,\,l_{18} =\frac{F_{28}^{0} F_{12}^{0} -F_{18}^{0} F_{22}^{0} }{\Delta },\,\,\,\,l_{21} =-\frac{F_{21}^{0} }{\Delta },\,\,\,l_{22} =\frac{F_{11}^{0} }{\Delta },\,\,\,\, \nonumber \\ l_{23}&=\frac{F_{11}^{1} F_{21}^{0} -F_{21}^{1} F_{11}^{0} }{\Delta },\,\,\,l_{24} =\frac{F_{12}^{1} F_{21}^{0} -F_{22}^{1} F_{11}^{0} }{\Delta },\,\,\,\,l_{25} =\frac{F_{15}^{0} F_{21}^{0} -F_{25}^{0} F_{11}^{0} }{\Delta },\,\,\,l_{31} =\frac{1}{F_{66}^{0} },\, \nonumber \\ l_{28}&=\frac{F_{18}^{0} F_{21}^{0} -F_{28}^{0} F_{11}^{0} }{\Delta },\,\,\,l_{32} =-\frac{2F_{66}^{1} }{F_{66}^{0} },\,\,l_{35} =\frac{F_{35}^{0} }{F_{66}^{0} },l_{38} =\frac{F_{38}^{0} }{F_{66}^{0} },\,\,\Delta =F_{11}^{0} F_{22}^{0} -F_{12}^{0} F_{21}^{0} ,\,\, \end{aligned}$$
(A.2)
in which
$$\begin{aligned} F_{1}^{q_{1} }&=\,\int \limits _{-h_{1} }^{-h_{2} } {Y_{11}^{(1)} (\bar{{z}})z^{q_{1} }\mathrm{dz}} +Y_{11}^{(2)} \int \limits _{-h_{2} }^{h_{3} } {z^{q_{1} }\mathrm{dz}} +\int \limits _{h_{3} }^{h_{4} } {Y_{11}^{(3)} (\bar{{z}})z^{q_{1} }\mathrm{dz}} , \nonumber \\ F_{2}^{q_{1} }&=\,\int \limits _{-h_{1} }^{-h_{2} } {Y_{12}^{(1)} (\bar{{z}})z^{q_{1} }\mathrm{dz}} +Y_{12}^{(2)} \int \limits _{-h_{2} }^{h_{3} } {z^{q_{1} }\mathrm{dz}} +\int \limits _{h_{3} }^{h_{4} } {Y_{12}^{(3)} (\bar{{z}})z^{q_{1} }\mathrm{dz}} , \nonumber \\ F_{6}^{q_{1} }&=\,\int \limits _{-h_{1} }^{-h_{2} } {Y_{66}^{(1)} (\bar{{z}})z^{q_{1} }\mathrm{dz}} +Y_{66}^{(2)} \int \limits _{-h_{2} }^{h_{3} } {z^{q_{1} }\mathrm{dz}} +\int \limits _{h_{3} }^{h_{4} } {Y_{66}^{(3)} (\bar{{z}})z^{q_{1} }\mathrm{dz}} ,\,\,q_{1} =0,\,1,\,2, \nonumber \\ F_{7}^{q_{2} }&=\int \limits _{-h_{1} }^{h_{2} } \;\;{{z}^{q_{2} }U_{1}^{(1)} Y_{11}^{(1)} \left( {\bar{{z}}} \right) \text{ d }z} +\int \limits _{-h_{2} }^{h_{3} } \; z^{q_{2} }U_{1}^{(2)} Y_{11}^{(2)} \left( {\bar{{z}}} \right) \mathrm{dz}+\int \limits _{h_{3} }^{h_{4} } \; \,z^{q_{2} }U_{1}^{(3)} Y_{11}^{(3)} \left( {\bar{{z}}} \right) \mathrm{dz}, \nonumber \\ F_{8}^{q_{2} }&=\int \limits _{-h_{1} }^{h_{2} } \;{{z}^{q_{2} }U_{2}^{(1)} Y_{12}^{(1)} \left( {\bar{{z}}} \right) \text{ d }z} +\int \limits _{-h_{2} }^{h_{3} } \; z^{q_{2} }U_{222}^{(2)} Y_{12}^{(2)} \left( {\bar{{z}}} \right) \mathrm{dz}+\int \limits _{h_{3} }^{h_{4} } \; \,z^{q_{2} }U_{2}^{(3)} Y_{12}^{(3)} \left( {\bar{{z}}} \right) \mathrm{dz}, \nonumber \\ F_{9}^{q_{2} }&=\int \limits _{-h_{1} }^{h_{2} }\;\; {\text{ z}^{q_{2} }U_{1}^{(1)} Y_{12}^{(1)} \left( {\bar{{z}}} \right) \text{ d }z} +\int \limits _{-h_{2} }^{h_{3} } z^{q_{2} }U_{1}^{(2)} Y_{12}^{(2)} \left( {\bar{{z}}} \right) \mathrm{dz}+\int \limits _{h_{3} }^{h_{4} } \,z^{q_{2} }U_{1}^{(3)} Y_{12}^{(3)} \left( {\bar{{z}}} \right) \mathrm{dz}, \nonumber \\ F_{10}^{q_{2} }&=\int \limits _{-h_{1} }^{h_{2} } { \text{ z}^{q_{2} }U_{2}^{(1)} Y_{11}^{(1)} \left( {\bar{{z}}} \right) \text{ d }z} +\int \limits _{-h_{2} }^{h_{3} } \; z^{q_{2} }U_{2}^{(2)} Y_{11}^{(2)} \left( {\bar{{z}}} \right) \mathrm{dz}+\int \limits _{h_{3} }^{h_{4} } \; \,z^{q_{2} }U_{2}^{(3)} Y_{11}^{(3)} \left( {\bar{{z}}} \right) \mathrm{dz}, \nonumber \\ F_{11}^{q_{2} }&=\int \limits _{-h_{1} }^{h_{2} } {\text{ z}^{q_{2} }U_{1}^{(1)} Y_{66}^{(1)} \left( {\bar{{z}}} \right) \text{ d }z} +\int \limits _{-h_{2} }^{h_{3} } \; z^{q_{2} }U_{1}^{(2)} Y_{66}^{(2)} \left( {\bar{{z}}} \right) \mathrm{dz}+\int \limits _{h_{3} }^{h_{4} } \; \,z^{q_{2} }U_{1}^{(3)} Y_{66}^{(3)} \left( {\bar{{z}}} \right) \mathrm{dz}, \nonumber \\ F_{12}^{q_{2} }&=\int \limits _{-h_{1} }^{h_{2} } { \text{ z}^{q_{2} }U_{2}^{(1)} Y_{66}^{(1)} \left( {\bar{{z}}} \right) \text{ d }z} +\int \limits _{-h_{2} }^{h_{3} } z^{q_{2} }U_{2}^{(2)} Y_{66}^{(2)} \left( {\bar{{z}}} \right) \mathrm{dz}+\int \limits _{h_{3} }^{h_{4} } \; \,z^{q_{2} }U_{2}^{(3)} Y_{66}^{(3)} \left( {\bar{{z}}} \right) \mathrm{dz},\,q_{2} =0,\,1.\, \end{aligned}$$
(A.3)
Appendix B
The parameters \(A_{lq} (l,q=1,2,3,4)\) are given as,
$$\begin{aligned} A_{11}&=\frac{\alpha _{1}^{2} \nabla _{b-1/2} }{4t_{2}^{3} }\left\{ {r_{12} \left[ {\alpha _{1}^{4} -3\left( {b-1} \right) \left( {b+1} \right) ^{3}-2\alpha _{1}^{2} \left( {b+4} \right) \left( {b+1} \right) } \right] } \right. \left. {+\left( {r_{11} -r_{31} } \right) \alpha _{2}^{2} \left( {b^{2}-b-2+\alpha _{1}^{2} } \right) } \right\} , \nonumber \\&\quad -\frac{\alpha _{1}^{2} \nabla _{b-1/2} }{8t_{2}^{3} }\left\{ {3\left( {2r_{31} -3r_{11} +r_{21} } \right) \alpha _{2}^{2} +\left( {r_{11} -5r_{12} -r_{22} } \right) \left[ {\left( {b+1} \right) ^{2}\left( {4b-5} \right) +\alpha _{1}^{2} \left( {4b+7} \right) } \right] } \right. , \nonumber \\&\quad \left. {+2\left( {7r_{12} +4r_{22} -4r_{11} -r_{21} } \right) \left[ {\left( {b^{2}-b-2} \right) +\alpha _{1}^{2} } \right] -9\left( {r_{11} -r_{12} -r_{22} +r_{21} } \right) } \right\} , \nonumber \\ A_{12}&=-\frac{\alpha _{1}^{2} \nabla _{b-1} }{4t_{2}^{4} }\left\{ {r_{13} \left[ {\left( {4-3b} \right) b^{3}-\left[ {2b\left( {b+2} \right) -\alpha _{1}^{2} } \right] \alpha _{1}^{2} } \right] } \right. +(r_{14} +r_{32} )\alpha _{2}^{2} \left[ {b\left( {b-2} \right) +\alpha _{1}^{2} } \right] , \nonumber \\&\quad +\left( {4r_{14} +4r_{32} +r_{24} } \right) \alpha _{2}^{2} +\left( {r_{23} +5r_{13} -r_{14} } \right) \left( {2b^{3}+2b\alpha _{1}^{2} -3b^{2}+\alpha _{1}^{2} } \right) , \nonumber \\&\quad \left. {+\left( {4r_{14} -4r_{23} -7r_{13} +r_{24} } \right) \left[ {b\left( {b-2} \right) +\alpha _{1}^{2} } \right] -3(r_{14} +r_{24} +r_{32} )\alpha _{2}^{2} -3\left( {r_{23} +r_{13} -r_{14} -r_{24} } \right) } \right\} , \nonumber \\ A_{13}&=\frac{\alpha _{1} \nabla _{b-1/2} }{8t_{2}^{3} }\left\{ {r_{35} \left[ {\left( {2b-1} \right) b+2\alpha _{1}^{2} } \right] \alpha _{2}^{2} } \right. -r_{15} \left[ {\left( {2b-1} \right) b^{3}+\alpha _{1}^{2} (3b-2\alpha _{1}^{2} )} \right] , \nonumber \\&\quad +(2r_{15} +r_{25} )\left[ {\alpha _{1}^{2} (1+2b)-b^{2}(1-2b)} \right] \,\left. {-2r_{25} \left[ {\left( {2b-1} \right) b+2\alpha _{1}^{2} } \right] } \right\} , \nonumber \\&\quad -\frac{\alpha _{1} \nabla _{b-1/2} }{8t_{2}^{3} }\left\{ {-U_{3} \left[ {b\left( {1+2b} \right) +2\alpha _{1}^{2} } \right] x_{2}^{2} +r_{35} \left( {2b+1} \right) \alpha _{2}^{2} } \right\} , \nonumber \\ A_{{14}}&=\frac{\alpha _{2} \alpha _{1}^{2} \nabla _{b-1/2} }{8t_{2}^{3} }\left\{ {1-2\left( {r_{38} +r_{18} } \right) \left[ {\left( {b-1} \right) b+\alpha _{1}^{2} } \right] +3r_{28} -2r_{18} -2r_{38} \,} \right\} , \nonumber \\ A_{{21}}&=\frac{\alpha _{2}^{2} \alpha _{1}^{2} \nabla _{b} }{4t_{2}^{2} }\left[ {r_{21} \alpha _{2}^{2} +\left( {r_{22} -r_{31} } \right) \left( {b^{2}+\alpha _{1}^{2} -1} \right) -r_{31} +r_{22} -r_{21} } \right] , \nonumber \\ A_{{22}}&=\frac{\alpha _{1}^{2} \alpha _{2}^{2} \nabla _{b-1/2} }{8t_{2}^{3} }\left\{ {2r_{24} \alpha _{2}^{2} +r_{32} +r_{23} -r_{24} 2(r_{32} +r_{23} )\left[ {(b-1)b+\alpha _{1}^{2} } \right] } \right\} , \nonumber \\ A_{{23}}&=\frac{\alpha _{1} \alpha _{2}^{2} }{4t_{2}^{2} }\left\{ {br_{35} \nabla _{b} +\frac{\left( {r_{25} +r_{35} } \right) \left[ {\left( {2b-1} \right) b+2\alpha _{1}^{2} } \right] \nabla _{b-1/2} }{2t_{2} }} \right\} , \nonumber \\ A_{{24}}&=-\frac{\alpha _{1}^{2} \alpha _{2} }{4}\left\{ {U_{4} \nabla _{b+1} +\frac{\left[ {r_{38} \left( {\alpha _{1}^{2} +b^{2}} \right) +r_{28} \alpha _{2}^{2} } \right] \nabla _{b} }{t_{2}^{2} }} \right\} , \nonumber \\ A_{{31}}&=\frac{\alpha _{1}^{2} \nabla _{b} }{4t_{2}^{3} }\left\{ {\alpha _{2}^{4} l_{11} +\alpha _{2}^{2} \left( {l_{31} +l_{21} +l_{12} } \right) \left( {b^{2}+\alpha _{1}^{2} -1} \right) +\alpha _{2}^{2} (2l_{31} +3l_{21} +l_{12} )} \right. \, \nonumber \\&\quad -\alpha _{2}^{2} \left( {l_{31} +2l_{21} +2l_{11} } \right) +l_{22} \left[ {\alpha _{1}^{4} -\left( {b+1} \right) ^{3}\left( {3b-1} \right) -2\left( {b+3} \right) \left( {b+1} \right) \alpha _{1}^{2} } \right] \nonumber \\&\quad +\left( {4l_{22} -l_{21} +l_{12} } \right) \left[ {\alpha _{1}^{2} (2b+3)+b^{2}(3+2b)-1} \right] \nonumber \\&\quad \left. {-\left( {5l_{22} +3l_{12} -3l_{21} -l_{11} } \right) \left( {\alpha _{1}^{2} +b^{2}-1} \right) +2\left( {l_{11} +l_{21} -l_{22} -l_{12} } \right) } \right\} , \nonumber \\ A_{{32}}&=\frac{\alpha _{1}^{2} \nabla _{b-1/2} }{8t_{2}^{4} }\left\{ {2\alpha _{2}^{4} l_{14} -2\left( {l_{32} -l_{13} -l_{24} } \right) \left( {b^{2}-b+\alpha _{1}^{2} } \right) } \right. +\alpha _{2}^{2} \left( {l_{13} -2l_{32} +3l_{24} } \right) \nonumber \\&\quad +2\alpha _{2}^{2} \left( {l_{32} -2l_{24} -2l_{14} } \right) +2l_{23} \left[ {\left( {2-3b} \right) b^{3}-2\alpha _{1}^{2} b\left( {b+1} \right) +\alpha _{1}^{4} } \right] \nonumber \\&\quad +\left( {l_{13} -l_{24} +4l_{23} } \right) \left( {4b^{3}-3b^{2}+4\alpha _{1}^{2} b+\alpha _{1}^{2} } \right) +2\left( {l_{14} -3l_{13} +3l_{24} -5l_{23} } \right) \left( {b^{2}-b+\alpha _{1}^{2} } \right) \nonumber \\&\quad \left. {-\left( {l_{14} -3l_{13} +3l_{24} -5l_{23} } \right) } \right\} \,+\frac{\alpha _{1}^{2} (b^{2}+\alpha _{1}^{2} )\nabla _{b} \cot \beta }{4t_{2}^{3} }, \nonumber \\ A_{{33}}&=\frac{(b^{2}+\alpha _{1}^{2} )\alpha _{1} \nabla _{b} }{4x_{2}^{3} }\left[ {\alpha _{2}^{2} \left( {l_{35} +l_{15} } \right) -l_{25} \left( {b^{2}-\alpha _{1}^{2} } \right) -\left( {l_{25} +l_{15} } \right) b-l_{15} } \right] , \nonumber \\ A_{{34}}&=\frac{\alpha _{1}^{2} \alpha _{2} \nabla _{b} }{4t_{2}^{3} }\left[ {l_{18} -l_{18} \alpha _{2}^{2} -\left( {l_{38} +l_{28} } \right) \left( {\alpha _{1}^{2} +b^{2}} \right) } \right] , \nonumber \\ A_{41}&=-\frac{0.25\left( {b^{2}+\alpha _{1}^{2} } \right) \alpha _{1}^{2} \nabla _{b} \cot \beta }{t_{2}^{2} },\,\,A_{42} =-T_{1} A_{T_{1} } -T_{2} A_{T_{2} }, \nonumber \\ A_{43}&=-\frac{0.25\alpha _{1} \nabla _{b+1/2} }{t_{2} }\left\{ {U_{3} \left[ {\left( {b+1/2} \right) b+\alpha _{1}^{2} } \right] +U_{4} \left( {b+1/2} \right) } \right\} ,\,\,\,A_{44} =\frac{0.25U_{4} \alpha _{1}^{2} \alpha _{2} \nabla _{b+1/2} }{t_{2} }, \nonumber \\ A_{T_{1} }&=-\frac{0.125\alpha _{1}^{2} (2\alpha _{1}^{2} +2b^{2}+2b-1)\nabla _{b+1/2} }{t_{2} \cot \beta },\,\,\,A_{T_{2} } =-\frac{0.125\alpha _{1}^{2} \left( {2\alpha _{2}^{2} +1} \right) \nabla _{b+1/2} }{t_{2} \cot \beta },\,\, \nonumber \\ A_{H_{P} }&=A_{T_{1} } +A_{T_{2} } =-\frac{0.0625\alpha _{1}^{2} \left( {2\alpha _{1}^{2} +4\alpha _{2}^{2} +2b^{2}+2b+1} \right) \nabla _{b+1/2} }{t_{2} \cot \beta }, \nonumber \\ \Pi _{1}&=-\left| {\begin{array}{ccc} A_{12}&{} A_{13}&{} \,A_{14} \\ A_{22}&{} A_{23}&{} \,A_{24} \\ A_{32}&{} A_{33}&{} \,A_{34} \\ \end{array}} \right| ,\quad \Pi _{2} =\left| {\begin{array}{ccc} A_{11}&{} A_{13}&{} \,A_{14} \\ A_{21}&{} A_{23}&{} \,A_{24} \\ A_{31}&{} A_{33}&{} \,A_{34} \\ \end{array}} \right| ,\quad \Pi _{3} =-\left| {\begin{array}{ccc} A_{11}&{} A_{12}&{} A_{14} \\ A_{21}&{} A_{22}&{} A_{24} \\ A_{31}&{} A_{32}&{} \,A_{34} \\ \end{array}} \right| ,\quad \Pi _{4} =\left| {\begin{array}{ccc} A_{11}&{} A_{12}&{} A_{13} \\ A_{21}&{} A_{22}&{} A_{23} \\ A_{31}&{} A_{32}&{} A_{33} \\ \end{array}} \right| \nonumber \\ \end{aligned}$$
(B1)
where
$$\begin{aligned} \nabla _{b+q} =\frac{\left[ {1-e^{-2t_{0} (b+q)}} \right] }{\left[ {\left( {b+q} \right) ^{2}+\alpha _{1}^{2} } \right] \left( {b+q} \right) },\,\,q=-1.0;-1/2;\,\,0;\,1/2;1.0. \end{aligned}$$
(B2)
